The connectedness of arithmetic progressions in Furstenberg's, Golomb's, and Kirch's topologies

• Autorzy: Szczuka, P.
• Języki publikacji: EN

Abstrakty

EN

In this paper we examine the connectedness of arithmetic progressions in the following topologies: Furstenberg's topology on the set of integers, Golomb's topology D on the set of positive integers, and Kirch's topology D′ on the set of positive integers. Immediate consequences of these studies are theorems concerning the connectedness and the locally connectedness of the topologies D and D′ proved by S. Golomb in 1959 and A. M. Kirch in 1969.

Słowa kluczowe

EN

Furstenberg's topology; Golomb's topology; Kirch's topology; connectedness; locally connectedness; arithmetic progressions; prime numbers

• Czasopismo: Demonstratio Mathematica
• Rocznik: 2010
• Tom: Vol. 43, nr 4
• Strony: 899-909
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Szczuka, P.

• Kazimierz Wielki University pl. Weyssenhoffa 11 85-072 Bydgoszcz, Poland,

Bibliografia

• [1] K. A. Broughan, Adic topologies for the rational integers, Canad. J. Math. 55 (2003), 711–723.
• [2] M. Brown, A countable connected Hausdorff space, Bull. Amer. Math. Soc. 59 (1953), 367.
• [3] R. Engelking, General Topology, PWN, Warsaw, 1977.
• [4] H. Furstenberg, On the infinitude of primes, Amer. Math. Monthly 62 (1955), 353.
• [5] S. Golomb, A connected topology for the integers, Amer. Math. Monthly 66 (1959), 663–665.
• [6] A. M. Kirch, A countable, connected, locally connected Hausdorff space, Amer. Math. Monthly 76 (1969), 169–171.
• [7] W. J. LeVeque, Topics in Number Theory, Vol. I, II, Dower Publications Inc., New York 2002.